Update

That’s a Tough One!

What can you do when you are stumped? Too many students sit and stare at the page, waiting for inspiration to strike — and when the solution doesn’t crack their heads open and step out, fully formed, they complain: “Math is too hard!”

So this year I have given my Math Club students a couple of mini-posters to put up on the wall above their desk, or wherever they do their math homework. The first gives four questions to ask yourself as you think through a math problem, and the second is a list of problem-solving strategies.

How to Solve a Tough Problem

Ask yourself these 4 questions:

1. What do I know?

List the facts or information given in the problem.

Underline or circle any key words, such as factor, multiple, area, or perimeter.

Watch out for mixed units!

Express the facts in math symbols, if you can.

2. What do I want?

Describe the goal, what the problem is asking you to find.

Underline or circle any key words, such as sum, product, next, or not. (Small words are easy to miss!)

Express the goal in math symbols, if you can.

3. What can I do?

Combine the given facts. Can you get closer to the goal?

Try a tool from your Problem Solving Tool Box.

Do one little step at a time.

4. Does it make sense?

When you get an answer, always look back at the original problem one more time.

Does your answer make sense?

Do you have the correct units (inches, cm2, kg, etc.)?

Can you think of a way to confirm that your answer is right?

Problem Solving Tool Box

Draw a diagram or picture.

Act the problem out, step by step.

Make a systematic list, chart, or table.

Look for a pattern.

Simplify the problem.
(Try it with smaller numbers.)

Restate the problem in another way, or look for a related problem.

Think about “Before” and “After” situations.

Work backwards.

Guess and check.
(Try something and see if it works.)

Sharing the Fun

If you would like to download these handouts for your students, here are the files:

“Scaffolding” is a metaphor from building — it’s just a support structure. It’s the “Problem Solving Toolbox” that I’m mainly seeing as ‘scaffolding’ for solving challenging problems, that would not be used for simple exercises.

Part of my trouble is that what I think should be a mere exercise turns out to be a real problem for the students. The 4 steps in “How to solve…” should help the kids who just stare at their homework and don’t know where to begin. The “Toolbox” is for when they get thoroughly stumped. It is oriented toward elementary Math Olympiads or other challenge problems.

Hi, Cora!
This post was really about how to solve story problems, but yours is a good question. The difficulty with your equation is that you have two unknown quantities: t and h. You have a relationship between them, which would be enough to graph a curve of possible values, but you can’t narrow it down to any specific numbers without more information.

If you want a method that will work on almost any math problem, I don’t know if I can simplify it more than the 4-step method above. For certain types of problems, there are specific steps that work most of the time. For instance, arithmetic word problems can often be solved with the 8-Step Model Drawing method.

Being a Math teacher and a perpetual student of mathematics I am convinced that to be able to solve any Math problem the most important thing is to be able to visualize. Clearer the images faster you would be able to solve it. Thus students should be encouraged to sketch a rough diagram and put all the given data on it. This way at a glance they will know what is given, what needs to be found out and what are the possible ways of finding it.

Here is another wonderful math website on Faster Math that will help us instantly power-up the math muscle.

Normally, I would delete comments linking to hard-sell websites, but you are clearly persistent. And you have a good point — that many math problems are easier to visualize by sketching a rough diagram.

For the future, however, please notice that your name becomes a link to your website. You don’t need to include the spammy-sounding advertising paragraphs. People who like your comment will click through on your name to see what else you have to say.

A COMPANY THAT PRODUCES CALCULATORS HAS OBSERVED THAT THE COST OF PRODUCTION OF X CALCULATORS(in hundreds) IS GIVEN BY THE FUNCTION Y=-2x+20.ALSO THE SELLING PRICE HAS BEEN MODELED INTO THE Y=2x+4.THE MAXIMUM NUMBER OF CALCULATORS THAT CAN BE PRODUCED IS 400.
SOLVE AND PLOT BOTH ON THE GRAPH

If this problem is copied from your homework, I can see why you are confused. The problem is poorly worded. Not very realistic, either — that second equation certainly flunks Econ 101!

I think this is what your teacher wants you to do:

Plot both equations on your graph, with y representing money (dollars or whatever you use) and x showing the number of calculators produced, in hundreds.

Also, plot the line x = 4, because 400 is the maximum number of calculators your company can make.

Find where the lines meet.

The two slanty lines on your graph will meet at the point where the selling price exactly equals the cost of production. You can’t work there without going out of business, and you certainly can’t sell calculators for less than they cost to make. It would make more sense to look for where that second line (selling price) has a greater y value than the first (cost), and x is still less than or equal to your maximum production capacity — but the data given in your problem has no such solution.

The easiest way I know to solve something like this is with educated “Guess and Check” — also known as trial and error. The ones digits give you a big clue: Y Y = _Y, which is only true for 0, 1, 5, or 6. So you need to try the 2-digit numbers that end in these digits.

If necessary, you could further analyze the problem. The tens digit of your answer must equal the ONES digit of XY plus the TENS digit of Y squared. But I wouldn’t bother with that — I would just grab a calculator and try numbers.

i need to know a question there a 7 lillypads and 3 frogs on each side how do you get the three frogs on 1 side to the other and the other side frogs to the other side. the rules are you can only jump 1 space and over 1 frog but cant go back what is the least moves you can do it in????
anyone now please let me know thank you

I can’t help you with your homework, but if you really want to learn math, try this:

(1) Print out the pdf pages linked in the article above. Hang them on the wall by your desk, or wherever you work on math. When you get stumped on a problem, ask yourself those questions, and think carefully about your answers.

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